Ensemble Kalman inversion (EKI) [1]} is an
offline variant of ensemble Kalman filtering designed to
learn parameters in a general model, rather than states of
a dynamical system. EKI can be viewed as a derivative-free optimization algorithm. Given a set of data \(\mathbf {y}\) , it iteratively evolves an ensemble of parameter estimates
both so that they achieve consensus and
evolve toward the optimal parameter value \(\mathbf {\theta }^*\)  (close to \(\mathbf {\theta }^\dagger \)  if the quality/volume of the data
is good enough) that minimizes an objective function, in our case given by the misfit (REF ), possibly with inclusion of a regularization term. It
has great potential for use with chaotic or stochastic models due to
its ensemble-based, derivative-free approach for optimizing
parameters. Theoretical work shows that noisy continuous-time versions of EKI exhibit an averaging effect that skips over
fluctuations superimposed onto the ergodic averaged forward model [2]}. Empirical results suggest similar phenomena apply to EKI as implemented here, justifying the application to noisy forward model evaluations. Furthermore, the derivative-free approach scales well to high-dimensional parameter spaces, as evidenced by the use of Kalman filtering in numerical weather prediction, where billions of parameters characterizing atmospheric states are routinely estimated [3]}. This makes the
algorithm appealing for complex climate models. The algorithm is mathematically proven to find the optimizer,
within an initial, ensemble-dependent subspace,
for linear models [4]}. It is known to be effective for high-dimensional nonlinear models [1]}, [6]}, [7]}, such as the nonlinear map from parameters to data represented by the idealized GCM we use in our proof-of-concept here.
